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**2015/1030**

**Title:**

SECOND COORDINATE SEQUENCE OF MP-LRS OVER NONTRIVIAL GALOIS RING OF ODD CHARACTERISTIC

**Authors:**
Vadim N. Tsypyschev

**Abstract:**

We investigate a well-known way to construct pseudo-random sequences by separation p-adic coordinate sequences of linear recurrences over Galois ring. Commonly it is necessary to know rank estimations of separated sequences. In this article we describe divisors of the minimal polynomial of the second p-adic coordinate sequence of the linear recurrent sequence of maximal period/MP-LRS over non-trivial Galois ring of odd characteristic in dependence of the initial vector of this LRS. Also we describe polynomials divisible by that minimal polynomial in dependence of the initial vector of this LRS. As a corollary we get non-trivial upper and lower estimations for the rank of the second coordinate sequence of such MP-LRS which provides us by possibility to use it in pseudo-random generation. We say that the Galois ring is non-trivial, if it differs from Galois field and from quotient ring too. These results were worked out with participation of V.L.Kurakin as a supervisor. Author is very grateful to V.L.Kurakin for his participation in this work

**ePrint:**
https://eprint.iacr.org/2015/1030

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